The graph of a complex function $f: \mathbb{C} \rightarrow \mathbb{C}$ is a 3-dimensional object in a 4-dimensional space and thus hard to visualize, even when it's a smooth 3-dimensional surface.
A natural way to visualize it is by two graphs of two functions $r: \mathbb{R}^2\rightarrow \mathbb{R}, i: \mathbb{R}^2\rightarrow \mathbb{R}$ with $r(z) = \text{Re}(f(z)), i(z) = \text{Im}(f(z))$. These functions define - under appropriate circumstances - two 2-dimensional objects $\color{red}{S^f_r}$ and $\color{green}{S^f_i}$ in a 3-dimensional space and can be visualized in $\mathbb{R}^3$, at least when they are smooth 2-dimensional surfaces.
In general these two surfaces may or may not
- intersect with each other at some points in $\mathbb{R}^3$ giving a 1-dimensional object $\color{blue}{C^f}= \color{red}{S^f_r} \cap \color{green}{S^f_i}$ which lives in $\mathbb{R}^3$
- intersect with the plane $\mathbb{R}^2$ giving two 1-dimensional objects $\color{red}{C^f_r} = \color{red}{S^f_r} \cap \mathbb{R}^2$ and $\color{green}{C^f_i} = \color{green}{S^f_i} \cap \mathbb{R}^2$ which live in $\mathbb{R}^2$
These 1-dimensional objects can be straight lines (or sets of straight lines), circles (or sets of circles), arbitrary open or closed curves (or sets of those).
One thing is obvious: There are $z_0 \in \mathbb{C}$ with $f(z) = 0$ (i.e. $f$ has roots) iff $\color{red}{C^f_r} \cap \color{green}{C^f_i} \neq \emptyset$. Knowing that each complex polynomial has roots (the fundamental theorem of algebra), we know that $\color{red}{C^P_r} \cap \color{green}{C^P_i} \neq \emptyset$ for all polynomials $P$. (We know even more: $\color{red}{C^P_r} \cap \color{green}{C^P_i}$ is a point set of size less or equal the degree of $P$.)
Example 1:
$P(z) = z^2 -1,\ r(x,y) = x^2 - y^2 - 1 ,\ i(x,y) = 2xy$
$\color{red}{C^P_r} = \{ (x,y)\ |\ x^2 - y^2 = 1 \}$, $\color{green}{C^P_i} = \{ (x,y)\ |\ x = 0 \vee y = 0 \}$
$P(z) = 0 \leftrightarrow z = (1,0) \vee z = (-1,0)$
Example 2:
$P(z) = z^2 + 1, r(x,y) = x^2 - y^2 + 1 , i(x,y) = 2xy$
$\color{red}{C^P_r} = \{ (x,y)\ |\ x^2 - y^2 = -1 \}$, $\color{green}{C^P_i} = \{ (x,y)\ |\ x = 0 \vee y = 0 \}$
$P(z) = 0 \leftrightarrow z = (0,1) \vee z = (0,-1)$
What I'd like to know:
What can - beyond the fundamental theorem - be said about the shapes and the positions of $\color{red}{C^P_r}$ and $\color{green}{C^P_i}$ for polynomials $P$ in general terms (depending on the degree of $P$)? May (or even does) the fundamental theorem result from the characterizations of $\color{red}{C^P_r}$ and $\color{green}{C^P_i}$?
How may holomorphic functions (= analytic complex functions) be characterized in terms of $\color{blue}{C^f}, \color{red}{C^f_r}, \color{green}{C^f_i}$?
- Like this: "For a holomorphic function $f$ the set $\color{blue}{C^f}$ must be non-empty and such-and-such"?
- Like that: "When a curve $C$ is such-and-such there is a holomorphic function $f$ with $C = \color{blue}{C^f}$"?
- Might a holomorphic function $f$ possibly be fixed by its $\color{blue}{C^f}$ alone or by the pair $(\color{red}{C^f_r}, \color{green}{C^f_i})$?
For the learned mathematician the answers to these question may seem obvious ("How can you ask?"), for me they are not, sorry.
